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Physical Review A

Information entropy and the space of decoherence functions in generalized quantum theory

Christopher John Isham, Noah Linden

Published Jun 1, 1997• 16 citations
Publisher

Abstract

In standard quantum theory, the ideas of information entropy and of pure states are closely linked. States are represented by density matrices \ensuremath{\rho} on a Hilbert space and the information entropy - tr(\ensuremath{\rho} ln \ensuremath{\rho}) is minimized on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalized quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions, we demonstrate that there are no ``pure'' decoherence functions. The main content of the paper is a notion of information entropy in generalized quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information entropy suggests an intrinsic way of giving a consistent set selection criterion.

DOI
10.1103/physreva.55.4030

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In standard quantum theory, the ideas of information entropy and of pure states are closely linked. States are represented by density matrices \ensuremath{\rho} on a Hilbert space and the information entropy - tr(\ensuremath{\rho} ln \ensuremath{\rho}) is minimized on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalized quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions, we demonstrate that there are no ``pure'' decoherence functions. The main content of the paper is a notion of information entropy in generalized quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information entropy suggests an intrinsic way of giving a consistent set selection criterion.
In standard quantum theory, the ideas of information-entropy and of pure states are closely linked. States are represented by density matrices ρρρ on a Hilbert space and the information-entropy −tr(ρlog⁡ρ)-tr(ρ\logρ)−tr(ρlogρ) is minimised on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalised quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions we demonstrate that there are no `pure' decoherence functions. The main content of the paper is a new notion of information-entropy in generalised quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information-entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information-entropy suggests an intrinsic way of giving a consistent set selection criterion.
The geometry of decoherence in generalized "consistent histories" quantum theory is explored, revealing properties of the theory that are independent of any particular application of it. It is shown how the decoherence functional of a closed quantum system may be regarded as an Hermitian form on the space of linear operators on the Hilbert space of the system. This identification makes manifest a number of structural properties of decoherence functionals. For example, a bound on the maximum number of histories in a consistent set is determined. When the decoherence functional is positive -- as in conventional quantum mechanics -- it defines a semi-inner product on the space of history operators. This shows that consistent sets of histories are just orthogonal sets in this inner product. It further implies the existence in general of Cauchy-Schwarz and triangle inequalities for positive decoherence functionals. The geometrical significance of the ILS theorem classifying all possible decoherence functionals is illuminated, and a version of the ILS theorem for decoherence functionals on class operators is given. The class of history operators consistent according to a given decoherence functional is found, and, conversely, it is shown how to construct the decoherence functionals according to which a given set of histories is consistent. More generally, the "geometric" point of view here developed supplies a powerful unified language with which to solve problems in generalized quantum theory.
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=∑j=1n⟨βj∣A∣βj⟩tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } tr(A):=j=1∑n​⟨βj​∣A∣βj​⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=∑j=1nPj∣⟨aj⟨aj∣\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|ρ=j=1∑n​Pj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,∑j=1npj=1pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1pj∈R,pj​0,j=1∑n​pj​=1
The notion of entropy was introduced by Clausius in order to discuss the thermal behavior of physical systems. After the work of Clausius, Boltzmann, Gibbs and others, the beginning of the exact treatment of entropy was due to von Neumann and Shannon. Their starting point was very di erent, von Neumann was motivated by quantum mechanics and Shannon founded communication theory. The entropy of a state describing a physical system is a quantity expressing the uncertainty or randomness of the system. Shannon regarded this uncertainty attached to a physical system as the amount of information carried by the system, so that the entropy of a state can be read as the information carried by the state. His idea comes from the following consideration: If a physical system has a large uncertainty and one receives information on the system by some procedure, then the so obtained information is more valuable than that received from a system having less uncertainty. It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behavior of certain probabilities in the course of independent trials. First we will discuss this phenomena for nite quantum systems. By a nite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a nite quantum system is a nite dimensional C*algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is "tracial " in the sense that Tr ab = Tr ba (a; b 2 A) Every functional! on A is determined by a density operator D! 2 A in the form!(a) = Tr D! a (a 2 A): The entropy S(!) of a functional! is de ned by means of its density operator as S(!) = Tr (D!): This notion was introduced by von Neumann in 1927 and we term it von Neumann's entropy or shortly entropy (cf. [1]). In fact, von Neumann de ned the entropy such
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
In usual quantum theory, the information available about a quantum system is defined in terms of the density matrix describing it on a spacelike surface. This definition must be generalized for extensions of quantum theory which neither require, nor always permit, a notion of state on a spacelike surface. In particular, it must be generalized for the generalized quantum theories appropriate when spacetime geometry fluctuates quantum mechanically or when geometry is fixed but not foliable by spacelike surfaces. This paper introduces a four-dimensional notion of the information available about a quantum system's boundary conditions in the various sets of decohering, coarse-grained histories it may display. This spacetime notion of information coincides with the familiar one when quantum theory is formulable in terms of states on spacelike surfaces but generalizes this notion when it cannot be so formulated. The idea of spacetime information is applied in several contexts: When spacetime geometry is fixed the information available through alternatives restricted to a fixed spacetime region is defined. The information available through histories of alternatives of general operators is compared to that obtained from the more limited coarse grainings of sum-over-histories quantum mechanics that refer only to coordinates. The definition of information is considered in generalized quantum theories. We consider as specific examples time-neutral quantum mechanics with initial and final conditions, quantum theories with nonunitary evolution, and the generalized quantum frameworks appropriate for quantum spacetime. In such theories complete information about a quantum system is not necessarily available on any spacelike surface but must be searched for throughout spacetime. The information loss commonly associated with the ``evolution of pure states into mixed states'' in black hole evaporation is thus not in conflict with the principles of generalized quantum mechanics.